Exploring Roots of
Quadratics

Erin Mueller

The general quadratic has equation . All of the possible roots (real and imaginary) of this
polynomial can be found using the quadratic formula , in which the radicand can tell us which roots
are real and which are imaginary. LetÕs explore how the variable ÒbÓ changes
the polynomial and the resulting roots. Below is a graph of a quadratic where
ÒbÓ is the altering variable.

As you can see, the vertex is traveling along a path that looks
much like another parabola itself. The vertex travels through the bottom two
quadrants depending on the sign of ÒbÓ. If ÒbÓ is positive, the vertex travels
to the right and in this case, parabola will be on the positive side of the
x-axis. If ÒbÓ is negative, the vertex travels to the left and will be on the
negative side of the x-axis.

Now letÕs consider graphing the parabola in the x-b plane. This
means that our ÒbÓ value will become our dependent variable while our ÒxÓ value
remains the independent variable.

By solving the general parabola for ÒbÓ, we will obtain a rational
equation.

When c=1, our graph of this equation looks like this:

From the above graph, we can conclude that when c=1, the range of
the function includes all real numbers except when . We can also see that whenever our x-value is negative, our
y-value (our actual ÒbÓ) is also negative and vice versa when our x-value is
positive.

If we take any particular value of b, say b = 3, and overlay this
equation on the graph we add a line parallel to the x-axis. If it intersects
the curve in the x-b plane the intersection points correspond to the roots of
the original equation for that value of b. We have the following graph.

The graph above shows that when b=3 in our original quadratic
equation , the roots for this equation are approximately x=-.4 and
x=-2.6. From the above, we have now graphed our equation in Òx-b planeÓ. This
means that instead of looking at x=intercepts for our roots, we are now looking
at the intersections of our rational polynomial with the horizontal line b=3.

Now letÕs look at this graph in motion.

For each value of b we select, we get a
horizontal line. It is clear on a single graph that we get two negative real
roots of the original equation when b > 2, one negative real root when b =
2, no real roots for -2 < b < 2, One positive real root when b = -2, and
two positive real roots when b < -2.

Consider the case when c = - 1 rather than + 1.

As you can see, we now have one positive root at
approximately x=.1 and another at approximately x=-7.9.
LetÕs try other values of ÒcÓ.

How about c=0?

When c=0, we only have one real negative root.
Since we still have a quadratic, we must have two roots. In this case, one is
real and one is imaginary.